Get a password manager already, and let it just generate random passwords for you. Typing in passwords is so lame. :) If you are on macOS I highly recommend https://github.com/ravenac95/sudolikeaboss (and by extension 1Password).
That said, lastpass can work without any modifications to your local machine(i.e. it can work without any browser plugins even) tho it's not very fabulously integrated, it does work... Assuming of course they don't block access to the lastpass website and JS.